Question

How do you write an equation for a circle with Point A is (4, -2) Point B is (10,6) as diameters?

Answer 1

We must first find the length of the diameter and the center of the circle.

The center is an equal distance from all points inside a circle. Therefore, we can use the midpoint formula `((x_1 + x_2)/2, (y_1 + y_2)/2)` to find the center.

`((x_1 + x_2)/2, (y_1 + y_2)/2)`

`=((4 + 10)/2, (-2 + 6)/2)`

`=(7, 2)`

The center will therefore be at `(7, 2)`.

Now for the length of the diameter.

This can be found by using the distance theorem, a simple variation on pythagorean theorem.

`d = sqrt((x_2 - x_1)^2 + (y_ 2 - y_1)^2)`

`d = sqrt(6^2 + 8^2)`

`d = sqrt(100)`

`d = 10`

Hence, the diameter measures `10` units. Since the equation of the circle is of the form `(x - h)^2 + (y - k)^2 = r^2`, where `(h, k)` is the center and `r` is the radius, we need the radius, and not the diameter. The equation `d = 2r` shows the relationship between the diameter (d) and the radius (r).

Solving for r:

`r = d/2`

`r = 10/2`

`r = 5`

Now that we know our radius, we can substitute what we know into the equation of the circle, of the form mentioned above:

`(x - 7)^2 + (y - 2)^2 = 25`

Here is the graph of this relation (note: it's not a function, since every value of x is not only with one value of y)

Practice exercises:

1. Determine the equation of the circle who's diameter ends at the points `(-1, -4)` and `(3, -5)`.

`2.` Determine the equation of the following circle.

Hopefully this helps, and good luck!